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The infinite series of sine and cosine terms of frequencies $0, \omega_{0}, 2\omega_{0}, 3\omega_{0}, ....k\omega_{0}$is known as trigonometric Fourier series and can written as, $$\mathrm{x(t)=a_{0}+\sum_{n=1}^{\infty}a_{n}\:cos\:n\omega_{0} t+b_{n}\:sin\:n\omega_{0} t… (1)}$$Here, the constant $a_{0}, a_{n}$ and $b_{n}$ are called trigonometric Fourier series coefficients.Evaluation of a0To evaluate the coefficient $a_{0}$, we shall integrate the equation (1) on both sides over one period, i.e., $$\mathrm{\int_{t_{0}}^{(t_{0}+T)}x(t)\:dt=a_{0}\int_{t_{0}}^{(t_{0}+T)}dt+\int_{t_{0}}^{(t_{0}+T)}\left(\sum_{n=1}^{\infty}a_{n}\:cos\:n\omega_{0} t+b_{n}\:sin\:n\omega_{0} t\right)dt}$$$$\mathrm{\Rightarrow\:\int_{t_{0}}^{(t_{0}+T)}x(t)\:dt=a_{0}T+\sum_{n=1}^{\infty}a_{n}\int_{t_{0}}^{(t_{0}+T)}cos\:n\omega_{0} t\:dt+\sum_{n=1}^{\infty}b_{n}\int_{t_{0}}^{(t_{0}+T)}sin\:n\omega_{0} t\:dt… (2)}$$As we know that the net areas of sinusoids over complete periods are zero for any non-zero integer n and any time $t_{0}$. Therefore, $$\mathrm{\int_{t_{0}}^{(t_{0}+T)}cos\:n\omega_{0} t\:dt=0\:\:and\:\:\int_{t_{0}}^{(t_{0}+T)}sin\:n\omega_{0} t\:dt=0}$$Hence, from equation (2), we get, $$\mathrm{\int_{t_{0}}^{(t_{0}+T)}x(t)\:dt=a_{0}T}$$$$\mathrm{\therefore\:a_{0}=\frac{1}{T}\int_{t_{0}}^{(t_{0}+T)}x(t)\:dt… (3)}$$Using equation (3), ... Read More
What is Fourier Series?In the domain of engineering, most of the phenomena are periodic in nature such as the alternating current and voltage. These periodic functions could be analysed by resolving into their constituent components by a process called the Fourier series.Therefore, the Fourier series can be defined as under −“The representation of periodic signals over a certain interval of time in terms of linear combination of orthogonal functions (i.e., sine and cosine functions) is known as Fourier series.”The Fourier series is applicable only to the periodic signals i.e. the signals which repeat itself periodically over an interval from $(-\infty\:to\:\infty)$and ... Read More
The cosine form of Fourier series is the alternate form of the trigonometric Fourier series. The cosine form Fourier series is also known as polar form Fourier series or harmonic form Fourier series.The trigonometric Fourier series of a function x(t) contains sine and cosine terms of the same frequency. That is, $$\mathrm{x(t)=a_{0}+\sum_{n=1}^{\infty}a_{n}\:cos\:n\omega_{0} t+b_{n}\:sin\:n\omega_{0} t… (1)}$$Where, $$\mathrm{a_{0}=\frac{1}{T}\int_{t_{0}}^{(t_{0}+T)}x(t)\:dt}$$$$\mathrm{a_{n}=\frac{2}{T}\int_{t_{0}}^{(t_{0}+T)}x(t)\:cos\:n\omega_{0} t\:dt}$$$$\mathrm{b_{n}=\frac{2}{T}\int_{t_{0}}^{(t_{0}+T)}x(t)\:sin\:n\omega_{0} t\:dt}$$In equation (1), by multiplying the numerator and denominator of the sine and cosine terms with ($\sqrt{a_{n}^{2}+b_{n}^{2}}$), we get, $$\mathrm{x(t)=a_{0}+\sum_{n=1}^{\infty}\left( \sqrt{a_{n}^{2}+b_{n}^{2}}\right)\left( \frac{a_{n}}{\sqrt{a_{n}^{2}+b_{n}^{2}}}cos\:n\omega_{0} t+\frac{b_{n}}{\sqrt{a_{n}^{2}+b_{n}^{2}}}sin\:n\omega_{0} t\right)… (2)}$$Putting the values in the equation (2) as, $$\mathrm{a_{0}=A_{0}}$$$$\mathrm{\sqrt{a_{n}^{2}+b_{n}^{2}}=A_{n}… (3)}$$$$\mathrm{\frac{a_{n}}{\sqrt{a_{n}^{2}+b_{n}^{2}}}=cos\:\theta_{n}\:\:and\:\:\frac{b_{n}}{\sqrt{a_{n}^{2}+b_{n}^{2}}}=-sin\:\theta_{n}}$$We obtain, $$\mathrm{x(t)=A_{0}+\sum_{n=1}^{\infty}A_{n}(cos\:\theta_{n}\:cos\:n\omega_{0} t-sin\:\theta_{n}\:sin\:n\omega_{0} t)}$$$$\mathrm{\Rightarrow\:x(t)=A_{0}+\sum_{n=1}^{\infty}A_{n}\:cos(n\omega_{0} t+\theta_{n})… (4)}$$Where, $$\mathrm{\theta_{n}=-tan^{-1} \left(\frac{b_{n}}{a_{n}}\right)… (5)}$$The ... Read More
Quarter Wave SymmetryA periodic function $x(t)$ which has either odd symmetry or even symmetry along with the half wave symmetry is said to have quarter wave symmetry.Mathematically, a periodic function $x(t)$ is said to have quarter wave symmetry, if it satisfies the following condition −$$\mathrm{x(t)=x(-t)\:or\:x(t)=-x(-t)\:and\:x(t)=-x\left (t ± \frac{T}{2}\right )}$$Some examples of periodic functions having quarter wave symmetry are shown in Figure-1.The Fourier series coefficients for the function having quarter wave symmetry are evaluated as follows −Case I – When n is odd$$\mathrm{x(t)=-x(-t)\:and\:x(t)=-x\left (t ± \frac{T}{2}\right )}$$For this case, $$\mathrm{a_{0}=0\:\:and\:\:a_{n}=0}$$And, $$\mathrm{b_{n}=\frac{8}{T} \int_{0}^{T/4}x(t)\:sin\:n\omega_{0}\:t\:dt}$$Case II – When n is even$$\mathrm{x(t)=x(-t)\:and\:x(t)=-x\left (t ± \frac{T}{2}\right ... Read More
Fourier SeriesIf $x(t)$ is a periodic function with period $T$, then the continuous-time exponential Fourier series of the function is defined as, $$\mathrm{x(t)=\sum_{n=−\infty}^{\infty}C_{n}\:e^{jn\omega_{0} t}… (1)}$$Where, $C_{n}$ is the exponential Fourier series coefficient, which is given by, $$\mathrm{C_{n}=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}x(t)\:e^{-jn\omega_{0} t}\:dt… (2)}$$Parseval’s Theorem and Parseval’s IdentityLet $x_{1}(t)$ and $x_{2}(t)$ two complex periodic functions with period T and with Fourier series coefficients $C_{n}$ and $D_{n}$.If, $$\mathrm{x_{1}(t)\overset{FT}{\leftrightarrow}C_{n}}$$$$\mathrm{x_{2}(t)\overset{FT}{\leftrightarrow}D_{n}}$$Then, the Parseval’s theorem of continuous time Fourier series states that$$\mathrm{\frac{1}{T} \int_{t_{0}}^{t_{0}+T} x_{1}(t)\:x_{2}^{*}(t)\:dt =\sum_{n=−\infty}^{\infty} C_{n}\:D_{n}^{*}\:[for\:complex\: x_{1}(t)\: \& \: x_{2}(t)] … (3)}$$And the parseval’s identity of Fourier series states that, if$$\mathrm{x_{1}(t)=x_{1}(t)=x(t)}$$Then, $$\mathrm{\frac{1}{T}\int_{t_{0}}^{t_{0}+T}|x(t)|^{2}\:dt=\sum_{n=−\infty}^{\infty}|C_{n}|^{2}… (4)}$$Proof – Parseval’s theorem or Parseval’s relation or ... Read More
Importance of Wave symmetryIf a periodic signal $x(t)$ has some type of symmetry, then some of the trigonometric Fourier series coefficients may become zero and hence the calculation of the coefficients becomes simple.Odd or Rotation SymmetryWhen a periodic function $x(t)$ is antisymmetric about the vertical axis, then the function is said to have the odd symmetry or rotation symmetry.Mathematically, a function $x(t)$ is said to have odd symmetry, if$$\mathrm{x(t)=-x(-t)… (1)}$$Some functions having odd symmetry are shown in the figure. It is clear that the odd symmetric functions are always antisymmetrical about the vertical axis.ExplanationAs we know that any periodic signal ... Read More
Fourier TransformThe Fourier transform is defined as a transformation technique which transforms signals from the continuous-time domain to the corresponding frequency domain and vice-versa. In other words, the Fourier transform is a mathematical technique that transforms a function of time $x(t)$ to a function of frequency X(ω) and vice-versa.For a continuous-time function $x(t)$, the Fourier transform of $x(t)$ can be defined as$$\mathrm{X(ω)=\int_{−\infty}^{\infty}x(t)\:e^{-j\omega t}dt}$$Points about Fourier TransformThe Fourier transform can be applied for both periodic as well as aperiodic signals.The Fourier transform is extensively used in the analysis of LTI (linear time invariant) systems, cryptography, signal processing, signal analysis, etc.Fourier transform ... Read More
Importance of Wave SymmetryIf a periodic signal $x(t)$ has some type of symmetry, then some of the trigonometric Fourier series coefficients may become zero and hence the calculation of the coefficients becomes simple.Half Wave SymmetryA periodic function $x(t)$ is said to have half wave symmetry, if it satisfies the following condition −$$\mathrm{x(t)=-x\left ( t ± \frac{T}{2}\right )… (1)}$$Where, $T$ is the time period of the function.A periodic function $x(t)$ having half wave symmetry is shown in the figure. It can be seen that this function is neither purely even nor purely odd. For such type of functions, the DC component ... Read More
Exponential Fourier SeriesA periodic signal can be represented over a certain interval of time in terms of the linear combination of orthogonal functions. If these orthogonal functions are exponential functions, then it is called the exponential Fourier seriesFor any periodic signal 𝑥(𝑡), the exponential form of Fourier series is given by, $$\mathrm{X(t)=\sum_{n=-\infty}^{\infty}C_n e^{jn\omega_0t}\:\:\:...(1)}$$Where, 𝜔0 = 2𝜋⁄𝑇 is the angular frequency of the periodic function.Coefficients of Exponential Fourier SeriesIn order to evaluate the coefficients of the exponential series, we multiply both sides of the equation (1) by 𝑒−𝑗𝑚𝜔0𝑡 and integrate over one period, so we have, $$\mathrm{\int_{t_0}^{t_0+T}x(t)e^{-jm\omega_0t}dt=\int_{t_0}^{t_0+T}(\sum_{n=-\infty}^{\infty}C_ne^{jn\omega_0t})e^{-jm\omega_{0}t}dt}$$$$\mathrm{\Rightarrow\int_{t_0}^{t_0+T}x(t)e^{-jm\omega_0t}dt=\sum_{n=-\infty}^{\infty}C_n\int_{t_0}^{t_0+T}e^{jn\omega_0t}e^{-jm\omega_0t}dt}$$$$\mathrm{\because \int_{t_0}^{t_0+T}e^{jn\omega_0t}e^{-jm\omega_0t}dt=\begin{cases}0 & for\: m ... Read More
Fourier TransformThe Fourier transform of a continuous-time function 𝑥(𝑡) can be defined as, $$\mathrm{X(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt}$$Convolution Property of Fourier TransformStatement – The convolution of two signals in time domain is equivalent to the multiplication of their spectra in frequency domain. Therefore, if$$\mathrm{x_1(t)\overset{FT}{\leftrightarrow}X_1(\omega)\:and\:x_2(t)\overset{FT}{\leftrightarrow}X_2(\omega)}$$Then, according to time convolution property of Fourier transform, $$\mathrm{x_1(t)*x_2(t)\overset{FT}{\leftrightarrow}X_1(\omega)*X_2(\omega)}$$ProofThe convolution of two continuous time signals 𝑥1(𝑡) and 𝑥2(𝑡) is defined as, $$\mathrm{x_1(t)*x_2(t)=\int_{-\infty}^{\infty}x_1(\tau)x_2(t-\tau)d\tau}$$Now, from the definition of Fourier transform, we have, $$\mathrm{X(\omega)=F[x_1(t)*x_2(t)]=\int_{-\infty}^{\infty}[x_1(t)*x_2(t)]e^{-j \omega t}dt}$$$$\mathrm{\Rightarrow F[x_1(t)*x_2(t)]=\int_{-\infty}^{\infty}[\int_{-\infty}^{\infty}x_1(\tau)x_2(t-\tau)d\tau]e^{-j \omega t}dt }$$By interchanging the order of integration, we get, $$\mathrm{\Rightarrow F[x_1(t)*x_2(t)]=\int_{-\infty}^{\infty}x_1(\tau)[\int_{-\infty}^{\infty}x_{2}(t-\tau)e^{-j \omega t}dt]d\tau }$$By replacing (𝑡 − 𝜏) = 𝑢 in the second integration, ... Read More